The cheapest from Weeze to Amsterdam (using public transport) is to walk across the border from Weeze and catch a bus on the Dutch side to the nearest train station. It might even be the fastest, too.

The connection from The Netherlands to Weeze (and back) is not quite convenient. You should at least consider visiting towns on the German side of the border.

So the two (cheap) transport options seem to be to walk either to a Dutch bus stop or to Weeze train station:

Weeze train station

Tickets from the train station to Amsterdam typically seem to cost around 40 euros, but sometimes are available for 25 euros if you book early and are lucky timewise. Tickets can be booked on bahn.de. As far as I can tell there is currently a train replacement bus leaving from the train station, but I believe this temporary.

Dutch bus stop

Prices

From the Dutch bus stop (Bushalte Kasteellaan, Well L) you can either take a bus to Venray with Arriva or to Nijmegen with Breng costing 2.83 and 8.12 respectively. Note that without a Dutch OV chipkaart this price might be higher (unless you figure out a way to buy the ticket digitally, Breng works with the Tranzer app and Arriva with the Glimble app, but I didn't check prices in there).

From Venray the normal train price to Amsterdam is 24.89 . If you buy at the train station it's one euro extra, but you can use the NS app instead. If however you buy through Gaiyo and you travel outside rush hour (The off-peak hours are on weekdays from 00:00 - 06:30, between 09:00 - 16:00 and between 18:30 - 24:00 the next morning, plus all day at weekends) it's 10% off for 22.40, for a total of 25.23 euros.

From Nijmegen the normal train price to Amsterdam is 20.70. With the 10% off that works out to a total (including bus) of 26.75.

But: There are often (not right now) better discounts available for Dutch trains which are often fixed price regardless of the distance, which can always be found on treinreiziger.nl (use Google Translate).

Times

Times when the best connections happen can be found on 9292.nl from "Bushalte Kasteellaan, Well L" to "Amsterdam Centraal". Travel time is 2.5 to 3 hours.

My recommendation

Take the German connection if it is available for the 25 euros, especially if it's fully by train. Otherwise check treinreiziger.nl and 9292.nl for the best connection.

Weeze Airport is poorly connected to public transportation and often the savings by flying Ryan Air are easily eaten up by transport from/to the airport. Take a look at https://www.weeze.de/de/inhalt/oeffentlicher-personennahverkehr/ Note that the busses don't run at night and even during the day it's once an hour at best.

"Best" options often depends on the specific time of day and it what trade offs in convenience you are willing to make. "Cheapest" is clearly walking: it's free. It takes about 13 hours to Düsseldorf and 30 hours to Amsterdam.

You can look at train tickets for your specific times to both locations at www.bahn.de . Weeze airport to Amsterdam will run you around 25 Euros and takes about 3 and a half hours to 4 hours. These will include a lot of changes.

The website https://www.rome2rio.com suggests bus to Weeze train station, train to Kleve, bus to Nijmegen, and then train to Amsterdam. Travel time 3:36, plus waiting times. From Rotterdam, train to Breda, train to Nijmegen.

If you want to cram in Dusseldorf (I think most people would choose Cologne of the Ruhr cities), I think it would be better to do Rotterdam->Amsterdam-Dusseldorf.

One option that hasn't been mentioned is ridesharing (blablacar.de). You have to be quite flexible with time, cannot plan so long in advance, and will probably need a bus or train to get from the airport to some town where someone is able to pick you up so it won't be a great option for such a short trip but it is typically cheaper than busses and quite a bit cheaper than trains.

Check the app "BlaBlaCar", which is basically an app for one-time carpooling. Somebody who goes remotely that direction will make a small detour to pick you up, you participate in gas money and they'll drop you off near your desired area. I just checked, the price for your trip should around 7€. This might be the cheapest option, takes 2 hours, however as it's not public transport per se, it's not predictable if someone is actually going this way when you actually need it. Apparently it's not a much offered route, but you could be lucky.

Edit:

I just realized you had multiple questions besides the initial one: Your trip seems rather confusing and too much for such a short amount of time. No matter what means of transport you use, even if everything seems close, you will be stuck in trains a lot. I'd rather focus on one aspect, either you want to check out Amsterdam and Rotterdam, while skipping Dusseldorf. Or you check out the German part of that area and come back for another trip to the Netherlands. Two cities + travel will already be more than enough for 4 days and I'm sure you do not want to only see the central station of Rotterdam and Dusseldorf, but explore a bit. Both are beautiful cities that deserve more than half a day. If it's more of a checklist of places you wanted to be at, or in case you want to sacrifice sleep it's possible, the train only takes 2-3h from Rotterdam to Dusseldorf.

Regarding your last question: I used to live in that area for a few decades and especially in Essen for 7 years (funny, never expected to read that cities' name here). My recommendation is to not go there or Duisburg if you don't have specific destinations there! While they can have their own certain vibe, it is not worth to sacrifice your really limited time. Both of these cities are characterized by the glorious days of coal mining and blue collar workers, which resulted in cities that don't have much to offer if you don't know where to look (and what you are looking for). Dusseldorf and Cologne are really friendly to tourists, there is so much going on you're barely able to cover everything major in 2 days + the nightlife is way more inviting, if you plan on going out in the nights. Essen and Duisburg would be way more historic and industrial, rather than fancy, clean cities.

Again, if you already have a certain destination or atmosphere you're looking for - go for it! But if you just considered those cities because they are big: reconsider!

It depends what you want to see and visit, but the cities on the German side are certainly worth a visit. There is a lot of industrial heritage: a mining-museum (more then 1 possibly), railroad-museum, villa Hugel/Krupp museum as well as other historic and big city cultural sights and events. The Ruhr area has a tourist site: https://www.ruhr-tourismus.de/en/subjects/culture/museums.html And of course you can check lonely planet, tripadvisor, and so on. It really depends on what you want to see.

The nice thing about the Ruhr area is that it is all well connected by public traffic. I think you have to decide what you are most interested in, and decide if you want to go either to the Netherlands or to Germany. Amsterdam and Rotterdam are also well worth a visit and have a good train connection between them.